NSW Education Standards Authority
2020
HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics Extension 2
General
Instructions
•
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•
•
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Reading time – 10 minutes
Working time – 3 hours
Write using black pen
Calculators approved by NESA may be used
A reference sheet is provided at the back of this paper
For questions in Section II, show relevant mathematical reasoning
and/or calculations
Total marks:
100
Section I – 10 marks (pages 2–5)
• Attempt Questions 1–10
• Allow about 15 minutes for this section
Section II – 90 marks (pages 6–14)
• Attempt Questions 11–16
• Allow about 2 hours and 45 minutes for this section
1400
Section I
10 marks
Attempt Questions 1–10
Allow about 15 minutes for this section
Use the multiple-choice answer sheet for Questions 1–10.
1
2
3
What is the length of the vector −i + 18 j − 6k ?
A.
5
B.
19
C.
25
D.
361
Given that z = 3 + i is a root of z 2 + pz + q = 0, where p and q are real, what are the
values of p and q ?
A.
p = − 6 , q = 10
B.
p = − 6 , q = 10
C.
p = 6,
q = 10
D.
p = 6,
q = 10
⎛ 1⎞
⎛ − 2⎞
What is the Cartesian equation of the line r = ⎜ ⎟ + λ ⎜ ⎟ ?
⎝ 4⎠
⎝ 3⎠
A.
2y + x = 7
B.
y − 2 x = −5
C.
y + 2x = 5
D.
2 y − x = −1
–2–
4
The diagram shows the complex number z on the Argand diagram.
z
O
Which of the following diagrams best shows the position of
A.
B.
O
O
C.
D.
O
O
–3–
z2
?
|z|
5
A particle undergoing simple harmonic motion has a maximum acceleration of 6 m /s2
and a maximum velocity of 4 m /s.
What is the period of the motion?
A.
B.
C.
D.
6
2p
3
3p
4p
3
⌠
1
Which expression is equal to ⎮ 2
dx ?
⌡ x + 4x + 10
A.
B.
C.
D.
7
p
1
⎛ x + 2⎞
+c
tan −1 ⎜
⎝ 6 ⎟⎠
6
⎛ x + 2⎞
tan −1 ⎜
+c
⎝ 6 ⎟⎠
1
2 6
ln
ln
x+2− 6
+c
x+2+ 6
x+2− 6
+c
x+2+ 6
Consider the proposition:
‘If 2n − 1 is not prime, then n is not prime’.
Given that each of the following statements is true, which statement disproves the
proposition?
A.
25 − 1 is prime
B.
26 − 1 is divisible by 9
C.
27 − 1 is prime
D.
211 − 1 is divisible by 23
–4–
8
Consider the statement:
‘If n is even, then if n is a multiple of 3, then n is a multiple of 6’.
Which of the following is the negation of this statement?
9
A.
n is odd and n is not a multiple of 3 or 6.
B.
n is even and n is a multiple of 3 but not a multiple of 6.
C.
If n is even, then n is not a multiple of 3 and n is not a multiple of 6.
D.
If n is odd, then if n is not a multiple of 3 then n is not a multiple of 6.
What is the maximum value of eiθ − 2 + eiθ + 2 for 0 ≤ q ≤ 2p ?
A.
5
B.
4
C.
2 5
D.
10
2a
10
⌠
Which of the following is equal to ⎮ ƒ ( x ) dx ?
⌡0
a
A.
⌠
⎮ ƒ ( x ) − ƒ ( 2a − x ) dx
⌡0
B.
⌠
⎮ ƒ ( x ) + ƒ ( 2a − x ) dx
⌡0
C.
⌠
2 ⎮ ƒ ( x − a ) dx
⌡0
D.
⌠ 1
⎮ 2 ƒ ( 2x ) dx
⌡0
a
a
a
–5–
Section II
90 marks
Attempt Questions 11–16
Allow about 2 hours and 45 minutes for this section
Answer each question in the appropriate writing booklet. Extra writing booklets are available.
For questions in Section II, your responses should include relevant mathematical reasoning
and/or calculations.
Question 11 (16 marks) Use the Question 11 Writing Booklet
(a)
Consider the complex numbers w = −1 + 4 i and z = 2 − i .
(i)
Evaluate | w | .
1
(ii)
Evaluate w z̄ .
2
e
(b)
⌠
Use integration by parts to evaluate ⎮ x ln x dx .
⌡1
3
(c)
A particle starts at the origin with velocity 1 and acceleration given by
3
a = v2 + v ,
where v is the velocity of the particle.
Find an expression for x, the displacement of the particle, in terms of v.
(d)
Consider the two vectors u = −2i − j + 3k and v = pi + j + 2k .
3
For what values of p are u − v and u + v perpendicular?
(e)
Solve z2 + 3z + (3 – i) = 0, giving your answer(s) in the form a + bi , where a
and b are real.
–6–
4
Question 12 (14 marks) Use the Question 12 Writing Booklet
(a)
A 50-kilogram box is initially at rest. The box is pulled along the ground with a
force of 200 newtons at an angle of 30° to the horizontal. The box experiences
a resistive force of 0.3R newtons, where R is the normal force, as shown in the
diagram.
Take the acceleration g due to gravity to be 10 m /s2.
R
200
30°
0.3R
50g
(i)
By resolving the forces vertically, show that R = 400.
2
(ii)
Show that the net force horizontally is approximately 53.2 newtons.
2
(iii)
Find the velocity of the box after the first three seconds.
2
Question 12 continues on page 8
–7–
Question 12 (continued)
(b)
A particle is projected from the origin with initial velocity u m /s at an angle q
to the horizontal. The particle lands at x = R on the x-axis. The acceleration
⎛ 0⎞
vector is given by a = ⎜ ⎟ , where g is the acceleration due to gravity. (Do
⎝ −g⎠
NOT prove this.)
y
u
q
R
(i)
x
Show that the position vector r(t) of the particle is given by
3
⎞
⎛ ut cosθ
⎟.
r(t) = ⎜
1
⎜⎝ ut sin θ − gt 2 ⎟⎠
2
(ii)
Show that the Cartesian equation of the path of flight is given by
y=
(iii)
3
⎞
−gx 2 ⎛
2u 2
2
θ
−
tan θ + 1⎟ .
tan
⎜
2 ⎝
gx
⎠
2u
Given u 2 > gR, prove that there are 2 distinct values of q for which the
particle will land at x = R.
End of Question 12
–8–
2
Question 13 (15 marks) Use the Question 13 Writing Booklet
(a)
p
. The central
3
point of motion of the particle is at x = 3 . When t = 0 the particle has its
A particle is undergoing simple harmonic motion with period
3
maximum displacement of 2 3 from the central point of motion.
Find an equation for the displacement, x, of the particle in terms of t.
(b)
Consider the two lines in three dimensions given by
3
⎛ 1⎞
⎛ 3⎞
⎛ −2⎞
⎛ 3⎞
⎜
⎟
⎜
⎟
⎜
⎟
r = −1 + λ1 2 and r = −6 + λ2 ⎜ 1 ⎟ .
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎜ ⎟
⎝ 7⎠
⎝ 1⎠
⎝ 2⎠
⎝ 3⎠
By equating components, find the point of intersection of the two lines.
(c)
(i)
By considering the right-angled triangle below, or otherwise,
a+b
prove that
≥ ab , where a > b ≥ 0.
2
a+b
2
a−b
x
(d)
(ii)
Prove that p 2 + 4 q 2 ≥ 4 pq .
1
(i)
Show that for any integer n, e inq + e −inq = 2 cos ( nq ) .
1
(ii)
By expanding e iq + e −iq , show that
(
cos4 q =
4
)
3
1
cos ( 4 q ) + 4 cos ( 2q ) + 3 .
8
(
)
π
(iii)
⌠2
Hence, or otherwise, find ⎮ cos4 θ dθ .
⌡0
–9–
2
Question 14 (16 marks) Use the Question 14 Writing Booklet
(a)
Let z1 be a complex number and let z2 =
iπ
e 3 z1 .
The diagram shows points A and B which represent z1 and z2 , respectively, in
the Argand plane.
( )
B z2
( )
A z1
O
(b)
(i)
Explain why triangle OAB is an equilateral triangle.
2
(ii)
Prove that z12 + z22 = z1 z2 .
3
A particle starts from rest and falls through a resisting medium so that its
acceleration, in m /s2, is modelled by
(
4
)
a = 10 1 − ( k v ) 2 ,
where v is the velocity of the particle in m /s and k = 0.01 .
Find the velocity of the particle after 5 seconds.
(c)
Prove by mathematical induction that, for n ≥ 2 ,
4
1
1
1 n −1
+ 2 + ··· + 2 <
.
2
2
3
n
n
(d)
Prove that for any integer n > 1, logn ( n + 1) is irrational.
– 10 –
3
Question 15 (13 marks) Use the Question 15 Writing Booklet
(a)
In the set of integers, let P be the proposition:
‘If k + 1 is divisible by 3, then k 3 + 1 is divisible by 3’.
(i)
Prove that the proposition P is true.
2
(ii)
Write down the contrapositive of the proposition P.
1
(iii)
Write down the converse of the proposition P and state, with reasons,
whether this converse is true or false.
3
Question 15 continues on page 12
– 11 –
Question 15 (continued)
(b)
CB m
= . The position vectors of A
AC n
and B are a and b respectively, as shown in the diagram.
The point C divides the interval AB so that
B
C
A
a
b
O
(i)
Show that AC =
n
(b − a).
m+n 2
(ii)
Prove that OC =
m
n
a+
b.
m+n m+n 1
Let OPQR be a parallelogram with OP = p and OR = r . The point S is the
midpoint of QR and T is the intersection of PR and OS, as shown in the diagram.
P
Q
p
T
S
O
r
R
(iii)
Show that OT =
2
1
r + p.
3 3
(iv)
Using parts (ii) and (iii), or otherwise, prove that T is the point that
divides the interval PR in the ratio 2 : 1.
3
End of Question 15
– 12 –
1
Question 16 (16 marks) Use the Question 16 Writing Booklet
(a)
Two masses, 2 m kg and 4 m kg, are attached by a light string. The string is
placed over a smooth pulley as shown.
The two masses are at rest before being released and v is the velocity of the
larger mass at time t seconds after they are released.
2m
v
4m
The force due to air resistance on each mass has magnitude k v, where k is a
positive constant.
(i)
(ii)
Show that
dv gm − kv
=
.
dt
3m
2
gm
3m
, show that when t =
ln 2 , the velocity of the
k
k
gm
larger mass is
.
2k
Given that v <
Question 16 continues on page 14
– 13 –
3
Question 16 (continued)
π
(b)
⌠2
Let In = ⎮ sin 2n+1 ( 2θ ) dθ , n = 0, 1, . . . .
⌡0
2n
I , n ≥ 1.
2n + 1 n − 1
(i)
Prove that In =
(ii)
22n (n!)2
Deduce that In =
.
( 2n + 1) !
3
3
1
⌠
Let Jn = ⎮ x n (1 − x )n dx , n = 0, 1, 2, . . . .
⌡0
(iii)
Using the result of part (ii), or otherwise, show that Jn =
(iv)
Prove that 2 n n !
(
2
) ≤ (2n + 1)! .
End of paper
– 14 –
(n!)2
( 2n + 1) !
.
3
2
BLANK PAGE
– 15 –
BLANK PAGE
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© 2020 NSW Education Standards Authority
NSW Education Standards Authority
2020
HIGHER SCHOOL CERTIFICATE EXAMINATION
Mathematics Advanced
Mathematics Extension 1
Mathematics Extension 2
1391
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–2–
–3–
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© 2020 NSW Education Standards Authority